# Cantellated 6-simplex

Cantellated 6-simplex
Rank6
TypeUniform
Notation
Bowers style acronymSril
Coxeter diagramx3o3x3o3o3o ()
Elements
Peta21 pentachoric prisms
7 rectified hexatera
7 small rhombated hexatera
Tera42 pentachora
105 tetrahedral prisms
42 rectified pentachora
21 small rhombated pentachora
Cells210 tetrahedra
210 triangular prisms
105 octahedra
35 cuboctahedra
Faces35+140+420 triangles, 210 squares
Edges105+420
Vertices105
Vertex figureTetrahedral pyramidal prism, edge lengths 1 (base tetrahedral prism and top edge) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {91}}{7}}\approx 1.36277}$
Hypervolume${\displaystyle {\frac {1181{\sqrt {7}}}{1920}}\approx 1.62741}$
Dipetal anglesRix–pen–penp: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{6}}\right)\approx 130.20297^{\circ }}$
Sarx–tepe–penp: ${\displaystyle \arccos \left(-{\frac {\sqrt {15}}{15}}\right)\approx 104.96322^{\circ }}$
Sarx–rap–rix: ${\displaystyle \arccos \left(-{\frac {1}{6}}\right)\approx 99.59407^{\circ }}$
Sarx–srip–sarx: ${\displaystyle \arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }}$
Central density1
Number of external pieces35
Level of complexity25
Related polytopes
ArmySril
RegimentSril
ConjugateNone
Abstract & topological properties
Flag count126000
Euler characteristic0
OrientableYes
Properties
SymmetryA6, order 5040
ConvexYes
NatureTame

The cantellated 6-simplex (also called the cantellated heptapeton, small rhombated heptapeton, or sril) is a convex uniform 6-polytope. It consists of 7 rectified hexatera, 7 small rhombated hexatera, and 21 pentachoric prisms as facets. 1 rectified hexateron, 4 small rhombated hexatera, and 2 pentachoric prisms join at each vertex. As the name suggests, it is the cantellation of the 6-simplex.

## Vertex coordinates

The vertices of a cantellated 6-simplex of edge length 1 can be given in seven dimensions as all permutations of:

• ${\displaystyle \left({\sqrt {2}},\,{\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,0,\,0,\,0,\,0\right)}$.

## Representations

A cantellated 6-simplex has the following Coxeter diagrams:

• x3o3x3o3o3o () (full symmetry)
• oxx3xxo3oox&#xt (A5 axial, rectified hexateron-first)

## Related polytopes

The cantellated 6-simplex is the colonel of a regiment of 31 members.